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A290339 Strong elliptic Carmichael numbers for the elliptic curve y^2 = x^3 + 80. 0
7783, 327781, 416521, 991681, 3401341, 4584187, 5234089, 5325883, 6966181, 8681581, 8787631, 10582501, 11947501, 16586089, 18143221, 18253777, 19687531, 22963471, 28478701, 55912891 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

An elliptic curve E over a field K is a nonsingular algebraic curve defined by a minimal Weierstrass equation

E/K: y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6

for some coefficients a_1, a_2, a_3, a_4, a_6 in K and discriminant of E not equal to zero.

Associated to E is an L-function L(E,s) = Sum_{N} a_N / N^s. The map sending the positive integer N to a_N is a multiplicative function. Moreover, a_p = p + 1 - #E(F_p) with E(F_p) defined below, and a_{p^e} = a_p a_{p^{e-1}} - 1_E(p) p a_{p^{e-2}} for all e >= 2 where 1_E(p) is 1 if p does not divide the discriminant of E and is 0 otherwise.

Let E(Q) be an elliptic curve over Q, the field of rational numbers. We can replace y and x with scalar multiples such that the defining equation of E has integer coefficients.

For an integer N > 2, let E(Z/NZ) be the set of points (x,y) satisfying the defining equation of E in Z/NZ, the ring of integers modulo N, and the "points at infinity" (identity element). If the discriminant of E is coprime to N, then E(Z/NZ) forms an Abelian group.

If the discriminant of E is indivisible by a prime p and if #E(F_p) = p, then p is called an anomalous prime for E. There are no anomalous primes for the curve y^2 = x^3 + 80. This was shown in a paper by H. Qin (2016), see link.

The notions of strong elliptic pseudoprimes and strong elliptic Carmichael numbers are defined in "Anomalous Primes and Extension of the Korselt Criterion", see link. A Korselt criterion for these two notions was proven in "Anomalous Primes and Extension of the Korselt Criterion" based on the Korselt criterion developed in "Elliptic Carmichael Numbers and Elliptic Korselt Criteria" (J. Silverman, 2012).

Let N be a composite number, and P be a point of E(Z/NZ). Suppose that N has at least two distinct prime factors and N is coprime to the discriminant of E. Furthermore, write N+1-a_N as N+1-a_N = 2^r * t for integers r and t with t odd. Then, N is a strong elliptic pseudoprime for (E,P) if tP is the identity or (2^s t) P is a point of the form (x,y) where 2y + a_1 x + a_3 = 0 modulo N for some integer s with 0 <= s < r. For a prime p, let ord_p(N) be the p-adic order of N. Also let e_{N,p}(E) be the exponent of the group E(Z/p^(ord_p(N))Z). N is a strong elliptic Carmichael number for E if and only if N has at least two distinct prime factors, N is coprime to the discriminant of E, and, for every prime p dividing N, e_{N,p}(E) divides t.

LINKS

Table of n, a(n) for n=1..20.

L. Babinkostova, J. C. Bahr, Y. H. Kim, E. Neyman, and G. K. Taylor, Anomalous primes and the Elliptic Korselt Criterion, arXiv:1608.02317 [math.NT], 2016.

L. Babinkostova, B. Bentz, M. I. Hassan, A. Hernández-Espiet and H. Kim, Anomalous Primes and Extension of the Korselt Criterion.

L. Babinkostova, B. Bentz, M. I. Hassan, A. Hernández-Espiet and H. Kim, Software for generating sequence.

D. M. Gordon, On the number of elliptic pseudoprimes, Mathematics of Computations Vol. 52:185 (1989), 231-245.

B. Mazur, Rational Points of Abelian Varieties with Values in Towers of Number Fields, Invent. Mathematics 18 (1972), 183-266.

I. Miyamoto and M. R. Murty, Elliptic Pseudoprimes, Mathematics of Computation, Vol. 53:187 (1989), 290-305.

S. Müller, On the existence and non-existence of elliptic pseudoprimes, Mathematics of Computation, Vol. 79 (2009), 1171-1190.

H. Qin, Anomalous primes of the elliptic curve E_D:y^2 = x^3 + D, Proceedings of London Mathematics Society, Vol. 3:112 (2016), 415-453.

J. H. Silverman, Elliptic Carmichael Numbers and Elliptic Korselt Criteria, Acta Arithmetica Vol. 155:3 (2012), 233-246.

EXAMPLE

Let N = 7783 = 43*181. The discriminant of E: y^2 = x^3 + 80 is -16*(4*0^3 + 27*80^2) = -2764800, which is coprime to N. It turns out that E(Z/43Z) is isomorphic to the Abelian group Z/7Z + Z/7Z and that E(Z/181Z) is isomorphic to the Abelian group Z/63Z + Z/3Z. In particular, #E(Z/43Z) = 49 and #E(Z/181Z) = 189, so a_43 = 43+1-49 = -5 and a_181 = 181+1-189 = -7. Therefore, a_N = a_43 * a_181 = 35, so N+1-a_N = 7749. Moreover, e_{N,43} = 7 and e_{N,181} = 63, both of which divide N+1-a_N.

CROSSREFS

Cf. A112927, A275739.

Sequence in context: A043520 A276144 A234032 * A234460 A253745 A253752

Adjacent sequences:  A290336 A290337 A290338 * A290340 A290341 A290342

KEYWORD

nonn

AUTHOR

Hyun Jong Kim, Jul 27 2017

STATUS

approved

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Last modified November 23 19:08 EST 2017. Contains 295128 sequences.